DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Tensor-invertible objects, fusion-ring units, and invertible operators

Definition

Let $C$ be a fusion category with unit $1$ . Its Grothendieck fusion ring $K_{0} (C)$ is the free abelian group on simple isomorphism classes, with multiplication induced by tensor product and coefficients given by the fusion rules of Fusion rules.

An object $X$ is tensor-invertible if there is an object $Y$ with $X \otimes Y ≅ 1$ and $Y \otimes X ≅ 1$ .
A class $[X]$ is a fusion-ring unit if it has a two-sided multiplicative inverse in $K_{0} (C)$ ; that inverse may have negative integer coefficients in the simple-object basis.
A linear operator is invertible only relative to a specified vector space $H$ and operator $T : H \to H$ : it must have a two-sided inverse $S : H \to H$ with $S T = T S = I_{H}$ .

These definitions concern different objects: categorical tensor product, a ring with virtual classes, and a chosen representation on a vector space.

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (standard reference, not scraped)
David Penneys, Unitary Fusion Categories, Math 8800 lecture notes (standard reference, not scraped)